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International Workshop on Algorithmic Learning Theory

ALT 1997: Algorithmic Learning Theory pp 323–338Cite as

Probability theory for the Brier game

  • Session 9
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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1316))

Abstract

The usual theory of prediction with expert advice does not differentiate between good and bad “experts”: its typical results only assert that it is possible to efficiently merge not too extensive pools of experts, no matter how good or how bad they are. On the other hand, it is natural to expect that good experts' predictions will in some way agree with the actual outcomes (e.g., they will be accurate on the average). In this paper we show that, in the case of the Brier prediction game (also known as the quadratic-loss game), the predictions of a good (in some weak and natural sense) expert must satisfy the law of large numbers (both strong and weak) and the law of the iterated logarithm; we also show that two good experts' predictions must be in asymptotic agreement. Finally, we briefly discuss possible extensions of our results to more general games; the limit theorems for sequences of events in conventional probability theory correspond to the log-loss game.

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Authors and Affiliations

  1. Department of Computer Science, University of London, Royal Holloway, TW20 0EX, Egham, Surrey, UK

    V. Vovk

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  1. V. Vovk
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Editor information

Ming Li Akira Maruoka

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© 1997 Springer-Verlag Berlin Heidelberg

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Vovk, V. (1997). Probability theory for the Brier game. In: Li, M., Maruoka, A. (eds) Algorithmic Learning Theory. ALT 1997. Lecture Notes in Computer Science, vol 1316. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63577-7_52

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  • DOI: https://doi.org/10.1007/3-540-63577-7_52

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63577-2

  • Online ISBN: 978-3-540-69602-5

  • eBook Packages: Springer Book Archive

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